##### Department of Mathematics,

University of California San Diego

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### Postdoc Seminar

## Marco Carfagnini

#### UCSD

## Small Fluctuations, Spectral Theory, and Random Geometry

##### Abstract:

The goal of this talk is to discuss new developments of random geometry. We will focus on small fluctuations (small balls) for degenerate diffusions and their connection to sub-Riemannian geometry. In particular, such diffusions can be used to describe spectral properties of their (hypoelliptic) generators, where the lack of ellipticity makes the analytic approach more challenging. Moreover, we will discuss random loops on Riemann surfaces which can be described in terms of SLEk loop measures. These are measures on the space of simple loops, and we will provide their asymptotics on small balls. Lastly, we will focus on the geometry of random Laplace eigenfunctions on the sphere and their application to physics and statistics.

### February 22, 2024

### 3:00 PM

APM 5829

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